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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 129360df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.hf3 | 129360df1 | \([0, 1, 0, -3495, 66660]\) | \(2508888064/396165\) | \(745734657360\) | \([2]\) | \(147456\) | \(1.0009\) | \(\Gamma_0(N)\)-optimal |
| 129360.hf2 | 129360df2 | \([0, 1, 0, -15500, -682452]\) | \(13674725584/1334025\) | \(40178357049600\) | \([2, 2]\) | \(294912\) | \(1.3475\) | |
| 129360.hf4 | 129360df3 | \([0, 1, 0, 18800, -3248092]\) | \(6099383804/41507235\) | \(-5000483523087360\) | \([4]\) | \(589824\) | \(1.6940\) | |
| 129360.hf1 | 129360df4 | \([0, 1, 0, -241880, -45867900]\) | \(12990838708516/144375\) | \(17393228160000\) | \([2]\) | \(589824\) | \(1.6940\) |
Rank
sage: E.rank()
The elliptic curves in class 129360df have rank \(1\).
Complex multiplication
The elliptic curves in class 129360df do not have complex multiplication.Modular form 129360.2.a.df
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
